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30 Jul 2005 - 05:44
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There really needs to be a "Carolyn Explains Everything For You" kind of page. I'm definitely going to want to look up some of these explanations down the road or just to re-read one if the idea escapes me (which it so often does.) -- SusanS - 30 Jul 2005
There really needs to be a "Carolyn Explains Everything For You" kind of page. I was just thinking exactly the same thing. Carolyn--do you want to add a "Carolyn Explains the World for You" topic thread?? But beyond that, I think a page would be good, too. I always find it helpful to be able to see the 'whole'--in this case, to be able to scroll down a page and see everything there. MathExplanationsFromCarolyn -- CatherineJohnson - 30 Jul 2005
I think my fundamentally retiring and humble nature renders my creating a "carolyn explains the world" topic thread an impossibility. Plus right now it seems I can't create new topic threads at all! I guess I broke something again (another problem to add to the list -- but most of the problems are actually physical ones, like pipes needing digging up in the basement, porches needing retiling, etc.). But how about we change that page title to "MathExplanationsFromCarolyn", and we'll all keep an eye out for stuff from posts to add to it? It also keeps duplication down if we copy only a couple of key lines and then put a link to the original explanation. Plus I would love feedback about how these explanations can be improved and made clearer and more useful. How about this one on dimensional analysis? If it's not crystal clear, I'd like to fix it. -- CarolynJohnston - 30 Jul 2005
OK, I like Carolyn explains the world to you, but I agree that 'math explanations from Carolyn' is a teensy bit less grandiose. Yup, I want to put in the dimensional analysis one; also one you wrote early on about multiplication (I'll check that one). -- CatherineJohnson - 30 Jul 2005
I have a question on this one. How are means, medians, & modes examples of functions that we can't express in an equation? I'm having trouble understanding how means, medians & modes are an instance of a machine that, given the same input, always produces the same output. -- CatherineJohnson - 30 Jul 2005
btw, something is wrong with the twiki in general, because I keep having it tell me it can't save....when in fact it has saved. Let's see, what else. hmm. Oh! You've got to get a link up to Jenny D! Also to The Education Wonks! -- CatherineJohnson - 30 Jul 2005
Means, medians, and modes are not examples of functions that can't be explained by formulas; they're not examples of functions at all. They are, rather, the tools we turn to in desperation when we have no means of finding a discernible formula. Or, more generally, that's the entire purpose of statistics. Let's give an example. Suppose I have a function in mind and you try to guess it. You give me 1 and I give you 23. You give me 2 and I give you 3. You give me 3 and I give you -7. You give me 4 and I give you 63. You give me 5 and I give you 5.23. What will I give you when you give me 6? See the pattern? No? Well, neither do I, because there isn't one! That's within the rules. All that is required is that I be consistent. If you try to trick me by giving me 1 again I am required to give you 23 again. Every time. That's the only requirement to make my replies a function. In real life we experience these kinds of patternless functions all the time, but we ignore them. For example, walk down the main street of your town and write down the heights of all the people you meet. (This isn't quite the usual definition of "function" because the inputs are people now, not numbers, but the idea is the same.) This is a function because if you run into the same person again they will continue to have the same height. It doesn't change. The same input (person) always gives the same output (height). (Not, strictly speaking, true, but close enough for the purposes of this problem--the height won't change significantly during the day of your experiment.) Now you have yourself a function, but can you discern any sort of pattern in it? In other words, can you predict what comes next? Surely you won't be able to make any reasonable guess about the next person you see. Or can you? Strictly speaking you can't, not in any precise sense, but you can still make a reasonable guess. Statistics offers us another completely different path to the Buddha, to wit, you can take the average of the people you've already measured, and you can use that as your next guess. You can guess that the next height you measure is probably going to be somewhat close to the mean. If you want to get fancy you could take the standard deviation of the people you have measured, guess that height has a Gaussian distribution, and reason that there is a 67% chance that the next person you will measure will be within one standard deviation of the mean you have measured. Pattern-based functions are useful because they offer us the magic elixir of being able to predict the future. Statistics gives us a rough means of predicting the future when standard reasoning breaks down. It is a useful proxy for functions when the function concept itself is of no practical value. -- WichitaBoy - 31 Jul 2005
Means, medians, and modes are not examples of functions that can't be explained by formulas; they're not examples of functions at all. They are, rather, the tools we turn to in desperation when we have no means of finding a discernible formula. Or, more generally, that's the entire purpose of statistics. I didn't think they were functions! That's a relief. Let me see why I thought that's what Carolyn said... Here it is:
They are, rather, the tools we turn to in desperation when we have no means of finding a discernible formula. Or, more generally, that's the entire purpose of statistics. Fantastic! I'm going to put this on the 'math explanations' page, too. This is a wonderful, concise explanation of WHY WE HAVE STATISTICS. This is exactly what I thought statistics was, btw, and I might have been able to put it this way, too (because of majoring in psychology as an undergrad, and then funding autism research for 7 years). -- CatherineJohnson - 31 Jul 2005
My whole adult life has been lived in the realm-of-no-formula. -- CatherineJohnson - 31 Jul 2005
All that is required is that I be consistent. If you try to trick me by giving me 1 again I am required to give you 23 again. Every time. That's the only requirement to make my replies a function. Hey! I know that function! That's the Bad Relationship Function! -- CatherineJohnson - 31 Jul 2005
Is there such a thing as a function that can't be described by a formula? Yes, WichitaBoy gave two: one was that weird one with f(5)=5.23, and the other was the relationship between people and their heights. The relationship between people and their heights is a more general example of a function than people are used to thinking about in school. Strictly speaking, a function has two properties: 1. A function is a "machine", or black box, that gives an output whenever it's given an input: 2. For every input, a function gives only a single output. Note that there's nothing in there that says that either the inputs or the outputs have to be numbers. In the height example that WichitaBoy gave, the outputs are numbers, but the inputs are individual people. Pretty clearly, there's no formula telling you what the next person's height is going to be, so it's an example of a function without a formula. Most datasets, in fact, are functions without a formula. Here's another example: the function that has as input an individual child, and as output the child's performance on a specific test. No formula. Here's another mind-stretching example: the function that has as input an individual, and as output the person's biological father. That's a function, because it has a unique output for every input -- everyone has a unique biological father (gene-splicing aside), but neither the input nor the output is a number. Needless to say, there's no formula. And here's another example, this time of a relationship that is not a function: if I created another input-output machine which took as input an individual and gave as output the person's sister, that would not be a function. Reasons: 1. Some people don't have sisters, and those people wouldn't have an output (rule 1 of a function says every input has to have an output). And even if we leave people without sisters out of the permissible inputs: 2. Some people have more than one sister. They might give you an output of "Clara" one time, and "Stephanie" another. So they break the only-one-output rule. However, the black box that takes an individual as input, and outputs the number of sisters they have, is a function. Everyone has some number of sisters from 0 through infinity, and everyone always has the same number of sisters at any given instant. If you're still listening, here's one important last thought: statistics typically handles "functions" where, however weird the inputs are, the outputs are numbers. If the outputs aren't numbers (as in the biological-father example), you can't use numerical methods to analyze the situation; you can't, for example, take the mean or standard deviation. -- CarolynJohnston - 31 Jul 2005
For every input, a function gives only a single output. OK, I need to slow down here and ACTUALLY READ YOUR POSTS ALL THE WAY THROUGH BEFORE PEPPERING YOU WITH QUESTION!! But since I'm too impulsive for that, let me ask one quick question: I understand that the output always has to be the same given the input, right? That's basic to a function. You guys are so great--this is definitely heading towards becoming a book. I'm wondering about the form of the book????? I wonder if the world is ready for a textbook specifically aimed at parents....or at adult 'learners' (I loathe that word) like me...... There's a gap out there, in the math LEARNER world. I think it may be related to your post on math books written before a field is mature. A person like me, who wants to learn LOTS more math RIGHT NOW (and there are quite a few of me, I think) can find self-teaching &/or review books that are quite good (I believe. I haven't used one yet, but I have no reason to think the people who have are wrong). BUT....there's no way for me to find the kind of book you were describing in your first post, and that's what I want. I want to learn more math, but I'd also like to 'learn' the excitement & the joy of math at my level which, as we know, is a seriously mature subject matter level. What's available to me is, in a sense, the worst of direct instruction. There is a huge focus on getting knowledge inside the self-teaching person's head, but that's it. There's no drama. Then there are books that are supposed to be fantastic (and I think probably are fantastic)--books like Sawyers' books, or maybe Martin Gardner's books. These are books whose intention is to convey the joy and beauty of mathematics to non-mathematicians without having them actually do any math. That's not what I want, either. I want a self-teaching textbook that conveys the drama and joy, AND gives me the kinds of mind-expanding problem sets the RUSSIAN MATH book has. -- CatherineJohnson - 31 Jul 2005
I want a self-teaching textbook that conveys the drama and joy, AND gives me the kinds of mind-expanding problem sets the RUSSIAN MATH book has. And I think that this sort of a book would really benefit teachers. Teachers are not stupid; they are adult learners. -- CarolynJohnston - 31 Jul 2005
I MUST READ THIS THREAD! -- CatherineJohnson - 31 Jul 2005
All this riffing on the definition of a function is interesting, but that's not the angle that jumped out at me from the original post. There are several aspects of the intelligent-verbal-games-in-the-car scenario that I really like.
Dan -- this is a good point -- we've done this sort of impromptu car stuff with all 3 kids, but I think it bears emphasizing. Car time is a great time to get fast facts down, and teach mental math. Everybody gets lots of praise for playing. -- CarolynJohnston - 01 Aug 2005
math talk in the car on the way home
We took the kids to a bar tonight, as it happened. Colin (17) is into playing the bass these days; he has a band that he plays with during the school year. I have a friend at work who is a hot guitar player and who just joined a classic rock band, and he was playing his first gig tonight, and they were letting kids stay through the first set, so we went to see him. It was a long drive for us -- all the way out to Greeley. The place was an authentic roadhouse with motorcycles parked out front, and the food was good -- it was Cajun food, and very authentic given that we were not in Cajun country but in Greeley, Colorado, home of the Feedlot You Can Smell All The Way To Denver. On the way home, Colin asked us about the difference between the median, the mean, and the mode of a data set, and what each of them is good for. This is, of course, the sort of thing we love to pontificate about. He then told us that he felt he had never really quite gotten the idea of a function, and asked us to explain it. It's a smart kid who understands what he doesn't understand. Most adults can't do that very well. Actually, most kids coming into calculus classes are confused by functions. A function is just a black box; you put in an input, and get out an output. What makes it a function is that, when you put in the same inputs, you always get the same outputs. You can't put the same number in the black box and get 2 one time, and 5 the next. Most texts teach functions using formulas to define the functions; all the functions kids see look like f(x)=3x-5, or g(x)=x/6. But functions don't have to have formulas to go with them; they can defy description by a formula. The only rule is that if you put in the same input multiple times, you get the same output, every time. The reason kids confuse formulas with functions is that it's hard to define functions that don't use formulas, even though in real life we encounter them all the time. When a function totally defies description with a formula, we often resort to trying to describe it with only a couple of numbers, such as the mean, median, and standard deviation (this is how the whole field of statistics arises). We played a 'figure-out-the-function' game on the way home from Greeley. Bernie and I would think of a function, and Colin and Ben would give us numbers for inputs, and we would then tell them the output. They'd then try to guess the formula we were using to define the function. They are both aces at extracting patterns. If anything, Ben would try to generalize from too little data; once he guessed, after one try, that the function was 'add 2'; he'd given me a 2, and I'd come back with 4 (the function I'd thought of was squaring; he got it on the next try). Bernie was giving Colin some functions that are so simple they trip up students with their obviousness, like the function that returns the same number you give it, and the one that returns '3' no matter what you give it. He gave Colin one function that was so bizarre you can't describe it with a pattern. Ben knew more about functions than I thought, even piping up with "that's the constant function 3" at the appropriate moment. Did they do functions one day for 5 minutes in Everyday Math? Well, he was definitely on the ball that day. Back to main page.Comments
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There really needs to be a "Carolyn Explains Everything For You" kind of page. I'm definitely going to want to look up some of these explanations down the road or just to re-read one if the idea escapes me (which it so often does.) -- SusanS - 30 Jul 2005
There really needs to be a "Carolyn Explains Everything For You" kind of page. I was just thinking exactly the same thing. Carolyn--do you want to add a "Carolyn Explains the World for You" topic thread?? But beyond that, I think a page would be good, too. I always find it helpful to be able to see the 'whole'--in this case, to be able to scroll down a page and see everything there. MathExplanationsFromCarolyn -- CatherineJohnson - 30 Jul 2005
I think my fundamentally retiring and humble nature renders my creating a "carolyn explains the world" topic thread an impossibility. Plus right now it seems I can't create new topic threads at all! I guess I broke something again (another problem to add to the list -- but most of the problems are actually physical ones, like pipes needing digging up in the basement, porches needing retiling, etc.). But how about we change that page title to "MathExplanationsFromCarolyn", and we'll all keep an eye out for stuff from posts to add to it? It also keeps duplication down if we copy only a couple of key lines and then put a link to the original explanation. Plus I would love feedback about how these explanations can be improved and made clearer and more useful. How about this one on dimensional analysis? If it's not crystal clear, I'd like to fix it. -- CarolynJohnston - 30 Jul 2005
OK, I like Carolyn explains the world to you, but I agree that 'math explanations from Carolyn' is a teensy bit less grandiose. Yup, I want to put in the dimensional analysis one; also one you wrote early on about multiplication (I'll check that one). -- CatherineJohnson - 30 Jul 2005
I have a question on this one. How are means, medians, & modes examples of functions that we can't express in an equation? I'm having trouble understanding how means, medians & modes are an instance of a machine that, given the same input, always produces the same output. -- CatherineJohnson - 30 Jul 2005
btw, something is wrong with the twiki in general, because I keep having it tell me it can't save....when in fact it has saved. Let's see, what else. hmm. Oh! You've got to get a link up to Jenny D! Also to The Education Wonks! -- CatherineJohnson - 30 Jul 2005
Means, medians, and modes are not examples of functions that can't be explained by formulas; they're not examples of functions at all. They are, rather, the tools we turn to in desperation when we have no means of finding a discernible formula. Or, more generally, that's the entire purpose of statistics. Let's give an example. Suppose I have a function in mind and you try to guess it. You give me 1 and I give you 23. You give me 2 and I give you 3. You give me 3 and I give you -7. You give me 4 and I give you 63. You give me 5 and I give you 5.23. What will I give you when you give me 6? See the pattern? No? Well, neither do I, because there isn't one! That's within the rules. All that is required is that I be consistent. If you try to trick me by giving me 1 again I am required to give you 23 again. Every time. That's the only requirement to make my replies a function. In real life we experience these kinds of patternless functions all the time, but we ignore them. For example, walk down the main street of your town and write down the heights of all the people you meet. (This isn't quite the usual definition of "function" because the inputs are people now, not numbers, but the idea is the same.) This is a function because if you run into the same person again they will continue to have the same height. It doesn't change. The same input (person) always gives the same output (height). (Not, strictly speaking, true, but close enough for the purposes of this problem--the height won't change significantly during the day of your experiment.) Now you have yourself a function, but can you discern any sort of pattern in it? In other words, can you predict what comes next? Surely you won't be able to make any reasonable guess about the next person you see. Or can you? Strictly speaking you can't, not in any precise sense, but you can still make a reasonable guess. Statistics offers us another completely different path to the Buddha, to wit, you can take the average of the people you've already measured, and you can use that as your next guess. You can guess that the next height you measure is probably going to be somewhat close to the mean. If you want to get fancy you could take the standard deviation of the people you have measured, guess that height has a Gaussian distribution, and reason that there is a 67% chance that the next person you will measure will be within one standard deviation of the mean you have measured. Pattern-based functions are useful because they offer us the magic elixir of being able to predict the future. Statistics gives us a rough means of predicting the future when standard reasoning breaks down. It is a useful proxy for functions when the function concept itself is of no practical value. -- WichitaBoy - 31 Jul 2005
Means, medians, and modes are not examples of functions that can't be explained by formulas; they're not examples of functions at all. They are, rather, the tools we turn to in desperation when we have no means of finding a discernible formula. Or, more generally, that's the entire purpose of statistics. I didn't think they were functions! That's a relief. Let me see why I thought that's what Carolyn said... Here it is:
The reason kids confuse formulas with functions is that it's hard to define functions that don't use formulas, even though in real life we encounter them all the time. When a function totally defies description with a formula, we often resort to trying to describe it with only a couple of numbers, such as the mean, median, and standard deviation (this is how the whole field of statistics arises).Is there such a thing as a function that can't be described by a formula? I think that's what Carolyn said..... And what would be an example? -- CatherineJohnson - 31 Jul 2005
They are, rather, the tools we turn to in desperation when we have no means of finding a discernible formula. Or, more generally, that's the entire purpose of statistics. Fantastic! I'm going to put this on the 'math explanations' page, too. This is a wonderful, concise explanation of WHY WE HAVE STATISTICS. This is exactly what I thought statistics was, btw, and I might have been able to put it this way, too (because of majoring in psychology as an undergrad, and then funding autism research for 7 years). -- CatherineJohnson - 31 Jul 2005
My whole adult life has been lived in the realm-of-no-formula. -- CatherineJohnson - 31 Jul 2005
All that is required is that I be consistent. If you try to trick me by giving me 1 again I am required to give you 23 again. Every time. That's the only requirement to make my replies a function. Hey! I know that function! That's the Bad Relationship Function! -- CatherineJohnson - 31 Jul 2005
Is there such a thing as a function that can't be described by a formula? Yes, WichitaBoy gave two: one was that weird one with f(5)=5.23, and the other was the relationship between people and their heights. The relationship between people and their heights is a more general example of a function than people are used to thinking about in school. Strictly speaking, a function has two properties: 1. A function is a "machine", or black box, that gives an output whenever it's given an input: 2. For every input, a function gives only a single output. Note that there's nothing in there that says that either the inputs or the outputs have to be numbers. In the height example that WichitaBoy gave, the outputs are numbers, but the inputs are individual people. Pretty clearly, there's no formula telling you what the next person's height is going to be, so it's an example of a function without a formula. Most datasets, in fact, are functions without a formula. Here's another example: the function that has as input an individual child, and as output the child's performance on a specific test. No formula. Here's another mind-stretching example: the function that has as input an individual, and as output the person's biological father. That's a function, because it has a unique output for every input -- everyone has a unique biological father (gene-splicing aside), but neither the input nor the output is a number. Needless to say, there's no formula. And here's another example, this time of a relationship that is not a function: if I created another input-output machine which took as input an individual and gave as output the person's sister, that would not be a function. Reasons: 1. Some people don't have sisters, and those people wouldn't have an output (rule 1 of a function says every input has to have an output). And even if we leave people without sisters out of the permissible inputs: 2. Some people have more than one sister. They might give you an output of "Clara" one time, and "Stephanie" another. So they break the only-one-output rule. However, the black box that takes an individual as input, and outputs the number of sisters they have, is a function. Everyone has some number of sisters from 0 through infinity, and everyone always has the same number of sisters at any given instant. If you're still listening, here's one important last thought: statistics typically handles "functions" where, however weird the inputs are, the outputs are numbers. If the outputs aren't numbers (as in the biological-father example), you can't use numerical methods to analyze the situation; you can't, for example, take the mean or standard deviation. -- CarolynJohnston - 31 Jul 2005
For every input, a function gives only a single output. OK, I need to slow down here and ACTUALLY READ YOUR POSTS ALL THE WAY THROUGH BEFORE PEPPERING YOU WITH QUESTION!! But since I'm too impulsive for that, let me ask one quick question: I understand that the output always has to be the same given the input, right? That's basic to a function. You guys are so great--this is definitely heading towards becoming a book. I'm wondering about the form of the book????? I wonder if the world is ready for a textbook specifically aimed at parents....or at adult 'learners' (I loathe that word) like me...... There's a gap out there, in the math LEARNER world. I think it may be related to your post on math books written before a field is mature. A person like me, who wants to learn LOTS more math RIGHT NOW (and there are quite a few of me, I think) can find self-teaching &/or review books that are quite good (I believe. I haven't used one yet, but I have no reason to think the people who have are wrong). BUT....there's no way for me to find the kind of book you were describing in your first post, and that's what I want. I want to learn more math, but I'd also like to 'learn' the excitement & the joy of math at my level which, as we know, is a seriously mature subject matter level. What's available to me is, in a sense, the worst of direct instruction. There is a huge focus on getting knowledge inside the self-teaching person's head, but that's it. There's no drama. Then there are books that are supposed to be fantastic (and I think probably are fantastic)--books like Sawyers' books, or maybe Martin Gardner's books. These are books whose intention is to convey the joy and beauty of mathematics to non-mathematicians without having them actually do any math. That's not what I want, either. I want a self-teaching textbook that conveys the drama and joy, AND gives me the kinds of mind-expanding problem sets the RUSSIAN MATH book has. -- CatherineJohnson - 31 Jul 2005
I want a self-teaching textbook that conveys the drama and joy, AND gives me the kinds of mind-expanding problem sets the RUSSIAN MATH book has. And I think that this sort of a book would really benefit teachers. Teachers are not stupid; they are adult learners. -- CarolynJohnston - 31 Jul 2005
I MUST READ THIS THREAD! -- CatherineJohnson - 31 Jul 2005
All this riffing on the definition of a function is interesting, but that's not the angle that jumped out at me from the original post. There are several aspects of the intelligent-verbal-games-in-the-car scenario that I really like.
- It encourages kids to think of otherwise idle time as time to exercise their brains. My daughters now often ask for questions whenever we are standing in a line or waiting in the car while Mom runs into a store, etc.
- You can also teach the other sibling to handle the other kid's mistake appropriately. A little teasing in good fun is okay. Taunts and insults don't fly. Don't try to build yourself up by highlighting the mistakes of others.
- I happen to think that there's a lot of effectiveness in "drill and kill" repetitive learning. Here's a chance to practice it without the pressure of a formal lesson time or detracting from time spent on deeper teaching. It is otherwise idle time, so quiz the kids on state capitals.
- You can morph at any time from this kind of quizzing to a deeper discussion of one topic--or switch over to completely unrelated family conversation whenever you like.
Dan -- this is a good point -- we've done this sort of impromptu car stuff with all 3 kids, but I think it bears emphasizing. Car time is a great time to get fast facts down, and teach mental math. Everybody gets lots of praise for playing. -- CarolynJohnston - 01 Aug 2005
| WebLogForm | |
|---|---|
| Title: | math talk in the car on the way home |
| TopicType: | WebLog |
| SubjectArea: | GamesAndActivities, MathProblemHelpLine, MiddleSchoolMath, ParentsTeachingKids |
| LogDate: | 200507300142 |